A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)
Issue:
Volume 7, Issue 6, December 2018
Pages:
78-87
Received:
13 November 2018
Accepted:
4 December 2018
Published:
2 January 2019
DOI:
10.11648/j.pamj.20180706.11
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Abstract: Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.
Abstract: Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmo...
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An Empirical Research on Mathematical Case Development in Mathematical Teaching in Liberal Arts College
Issue:
Volume 7, Issue 6, December 2018
Pages:
88-94
Received:
15 November 2018
Accepted:
8 December 2018
Published:
2 January 2019
DOI:
10.11648/j.pamj.20180706.12
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Abstract: In order to explore mathematics learning methods suitable for liberal arts students specially and the educational value of mathematics application cases developed by liberal arts students, an experiment of mathematics application case study was carried out in the first grade liberal arts undergraduate students. 205 mathematics application cases have been obtained, and case studies were used with the statistical summarization and analytical generalization methods. From 2009 to 2013, about 531 liberal arts freshmen in statistical course completed a total of 90 mathematics survey reports containing the university campus life (58 cases), professional related research (21 cases), social issues (11 cases) and so on. From 2014 to 2018, liberal arts students completed 115 mathematics application cases in Applied Probability and Statistics class which can be divided into 3 categories according to the degree of innovation: using mathematical knowledge to solve problems (79 cases), empirical survey (14 cases) and literature analysis (22 cases). The case topics selected by liberal arts students are all related to their own interests, and the innovative uses are for the vast majority. Conclusion: (i) writing a mathematics survey report is an appropriate way for students to apply mathematics; (ii) mathematics application is the fusion of interest and knowledge in probability and statistical area for liberal arts undergraduate students; (iii) the development process of mathematics application cases can demonstrate innovative thinking and application ability for liberal arts undergraduate students.
Abstract: In order to explore mathematics learning methods suitable for liberal arts students specially and the educational value of mathematics application cases developed by liberal arts students, an experiment of mathematics application case study was carried out in the first grade liberal arts undergraduate students. 205 mathematics application cases hav...
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Post and Jablonsky Algebras of Compositions (Superpositions)
Issue:
Volume 7, Issue 6, December 2018
Pages:
95-100
Received:
29 November 2018
Accepted:
14 December 2018
Published:
10 January 2019
DOI:
10.11648/j.pamj.20180706.13
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Abstract: There are two algebras of compositions, Post and Jablonsky algebras. Definitions of these algebras was very simple. The article gives mathematically precise definition of these algebras by using Mal’cev’s definitions of the algebras. A. I. Mal’cev defined pre-iterative and iterative algebras of compositions. The significant extension of pre-iterative algebra is given in the article. Iterative algebra is incorrect. E. L. Post used implicitly pre-iterative algebra. S. V. Jablonsky used implicitly iterative algebra. The Jablonsky algebra has the operation of adding fictitious variables. But this operation is not primitive, since the addition of fictitious variables is possible at absence of this operation. If fictitious functions are deleted in the Jablonsky algebra then this algebra becomes correct. A natural classification of closed sets is given and fictitious closed sets are exposed. The number of fictitious closed sets is continual, the number of essential closed sets is countable.
Abstract: There are two algebras of compositions, Post and Jablonsky algebras. Definitions of these algebras was very simple. The article gives mathematically precise definition of these algebras by using Mal’cev’s definitions of the algebras. A. I. Mal’cev defined pre-iterative and iterative algebras of compositions. The significant extension of pre-iterati...
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